State-space models can be used to incorporate subject knowledge on the underlying dynamics of a time series by the introduction of a latent Markov state-process. A user can specify the dynamics of this process together with how the state relates to partial and noisy observations that have been made. Inference and prediction then involves solving a challenging inverse problem: calculating the conditional distribution of quantities of interest given the observations. This article reviews Monte Carlo algorithms for solving this inverse problem, covering methods based on the particle filter and the ensemble Kalman filter. We discuss the challenges posed by models with high-dimensional states, joint estimation of parameters and the state, and inference for the history of the state process. We also point out some potential new developments which will be important for tackling cutting-edge filtering applications.